Mathematical Proof that the exchange rate for BTC/USD will reach infinity

Let me preface this by saying nothing in the future is a guarantee. I am going to make a lot of assumptions in this post, but I don't personally think any are unreasonable. This is also a thought experiment - my word should in no way be taken as gospel. Well, here goes.

Assumptions:

1. The amount of supply Bitcoin in existence will never exceed 21m, or S
2. There is a subset of people in this world (like myself) who will exchange their fiat for Bitcoin at any price. We will call this demand from hodlers, or D^H. Since these people will buy at any price, we can assume D^H is a positive, changing value.
3. The value of D^H will grow in a direct relationship with inflation. It may not be proportional, but more dollars in market = more expendable income = more money hodlers put into BTC.
4. People will constantly trade into and out of Bitcoin aka people who want quick gains and do not want to use Bitcoin as their treasury reserve asset. We will call this demand from traders, or D^T.
5. Demand from traders can either be a negative or a positive value depending on time. However, as the number of hodlers in the market increase, the number of traders decreases. Therefore, I assume |D^T| will get closer to 0 overtime. Note that it will never truly be 0, but we can assume it to be close enough over a long period of time to assume this value to be insignificant.
6. The US dollar will always continue to depreciate in value (aka there will never be negative inflation). The inflation rate is measure as variable r and we can assume it is a fluctuating, positive value

Okay, lets start with our base function. Most of us know that prices in markets are set by 2 things: supply and demand. We will write our first function as:

P($) = f(S,D) = D/S

This ratio indicates that price, P in $, is a function of supply in demand. The function shows a direct relationship with demand (i.e. demand goes up, price goes up). It also shows an inverse relationship with supply (supply goes up, price goes down). This function is applicable to all markets.

Now, as mentioned in assumptions 2 and 4, I can assume demand is coming from 2 different types of people: hodlers and traders. I also know supply is at 21 million. I will rewrite my equation as:

P($) = f(D^H,D^T) = (D^H + D^T)/21,000,000

We are not done yet though because we know demand is a factor of time. As mentioned in assumption 3, D^H will adjust for inflation. Lets take todays demand from hodlers as our constant and call it D^H,CONST and say that D^H = D^H,CONST * i where i is inflation. New equation is:

P($) = f(i,D^T) = (D^H,CONST * i + D^T)/21,000,000

Inflation compounds on top of itself using compound interest formula. This is expressed as

i = (1 + r/n)^(nt)

where r is the rate of inflation month-to-month, t is the time period (years), and n is the number of times compounded (we will assume 12 because month-to-month inflation is reported in every issue of CPI. Replacing the values we see:

i = 1 + r/12)^(12t)

Plug this back into our main equation to yield

P($) = f(r,t,D^T) = (D^H,CONST * ((1 + r/12)^(12t)) + D^T)/21,000,000

We know r and D^T are not constants, they are also functions of time. However, as previously mentioned, r is always a positive value and D^T has an inverse relationship with time. Rewrite final equation as

P($) = f(r,t,D^T) = (D^H,CONST * ((1 + r(t)/12)^(12t)) + D^T(t))/21,000,000

Okay, now that we have our final equation to model price, lets take the limit as time approaches ∞

lim(t -> ∞) P($) = (D^H,CONST * ((1 + r(∞)/12)^(12*∞)) + D^T(∞))/21,000,000

We know that any positive real number great than 1 when taken the power of ∞ is ∞. We also know that 12*∞ is ∞. Now plug in and simplify

lim(t -> ∞) P($) = (D^H,CONST * ((x > 1)^∞) + 0)/21,000,000

lim(t -> ∞) P($) = (D^H,CONST * ∞+ 0)/21,000,000

lim(t -> ∞) P($) = (∞+0)/21,000,000

lim(t -> ∞) P($) = ∞/21,000,000

lim(t -> ∞) P($) = ∞

All of this to say if you buy and hodl, graph go up and to the right. Or as Saylor puts it, "it's going up forever, Laura".